Find the derivative $LaTeX: \displaystyle f(x) = \frac{- 5 x^{\frac{5}{2}} + 2 x^{\frac{3}{2}} + 5 x^{\frac{1}{2}}}{x^{\frac{1}{2}}}$

Using termwise division gives $LaTeX: \displaystyle f(x) = - 5 x^{2} + 2 x + 5$ . Now the power rule for derivatives can be used instead of the quotient rule. This gives LaTeX: f'(x) = 2 - 10 x